PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number I, September 1980 COUNTABLY COMPACT, LOCALLY COUNTABLE T2-SPACES J. E. VAUGHAN Abstract. The results of this paper provide a simple method for constructing locally countable 7"2-spaces (not T3) in which every infinite closed set has cardinal- ity 2C. The spaces are used in a variety of ways as counterexamples. One of these spaces may be considered as a countably compact version of the Katëtov //-closed extension of the natural numbers. 1. Introduction. The main result in this paper (Theorem 1.1) gives a method for constructing T2-spaces which have small local cardinality, but which also'satisfy the property that every infinite closed set has cardinality 2C. In Corollary 1.2, we specialize this result to the Stone-Cech compactification of the natural numbers ß(u>), and the resulting space, which may be considered as a countably compact version of the Katëtov extension of the natural numbers, gives rise to the following examples. (A) A countably compact, locally countable T2-space which is not D-compact for any D in /?(co)\«. (B) Locally countable T2-spaces P and Q in which every infinite closed set has cardinality 2C,and P X Q is not countably compact. (C) A space which shows that property wD is not hereditary in the class of Urysohn spaces in which every point is a Gs. (D) A T2-space in which every point is a Gs, but which has no countably infinite closed sets. (E) A countably compact, locally countable T2-space in which there are no nontrivial convergent sequences. (This strengthens an example of Charles Aull [1].) (F) A countably compact T2-space with density w and weight 2C. (This strengthens one use of the Katëtov extension of the natural numbers.) By applying Theorem 1.1 to the Gleason space of the closed unit interval we get (G) A countably compact, 77-closed, locally countable T2-space which is not a Baire space (this adds "locally countable" to an example of Eric van Douwen [4]). 2. Definitions and statements of results. In order to state our results and describe the examples, we need a number of definitions. We begin with those definitions needed for the statement of Theorem 1.1, and put the remaining definitions at the end of this section. Received by the editors June 27, 1979 and, in revised form, September 17, 1979; presented to the Ohio University Topology Conference, March 15, 1979. 1980 MathematicsSubject Classification.Primary 54D20, 54D25, 54D30, 54G20; Secondary 54D35, 54A25. Key words and phrases. Countably compact, locally countable, //-closed, property wD, Katëtov extension, right separated, Z)-compact, no nontrivial convergent sequences, Baire space. © 1980 American Mathematical Society 0002-99 39/80/0000-0426/$02.7S 147 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 148 J. E. VAUGHAN Let (X, S) denote a set X and a topology S on X. For A c X, let cls(A) denote the closure of A in the space (X, S). Let « denote the first infinite cardinal (i.e., the set of natural numbers) and c = 2". For a set E, let \E| denote the cardinal number of E. The local cardinality of (X, S) is the smallest cardinal k > to such that for every x G X, there exists U G S such that x G U and | U\ < k. For a dense set D g X, let SD denote the topology on X having as a base B = {{x} u (U n D): x G U G S}. We call 5fl the Katëtov extension of S with respect to D. Clearly (1) S c SD, (2) D G SD, and (3) X\D is a closed discrete set in (A', SD). The topology SD is used as an aid to understanding the topology T constructed in Theorem 1.1. Let (ß(u), S) be the Stone-Cech compactification of the natural numbers. Since u is dense in (ß(u), S) we may consider the topology Sa on ß(co). We denote this topology by K and note that ( ß(u>), K) is the Katëtov extension of the natural numbers w. A space is called right separated provided there exists a well-order on X such that every initial segment in the well-order is an open set in the space. We now can state 1.1 Theorem. Let (X, S) be a T2-space in which every infinite closed set has cardinality 2C,and let D be a dense subset of (X, S) with \D\ < \X\ = 2C. Then there exists a topology T on X satisfying the following conditions: A. S G T G SD. B. The local cardinality of(X, T) is < \D\. C. Every infinite closed subset of(X, T) has cardinality 2C. D. There is a discrete subspace Z of (X, T) such that for every infinite set F G X, we have \clT(F) - Z\ = 2C,and further if F c D, then |clr(F) n Z| = 2C. E. If D with the subspace topology induced from S is right separated, then (X, T) is right separated. Our main use of Theorem 1.1 is the following result. We have included in the statement the values of most of the basic cardinal functions for the topology T. These are all easily calculated and left to the reader. 1.2 Corollary. Let (ß(u>), S) denote the Stone-Cech compactification of a and (/?(«), K) the Katëtov H-closed extension of w. There exists a topology T on ß(u) such that the following hold: A. S G T G K. Thus ( ß(u), T) is a Urysohn space, hence strongly Hausdorff in the sense of [8], and a T2-space. Each n G w is isolated in ( /8(to), T) and w is dense in (ß(w), T). Therefore, the density, cellularity, m-weight, and depth of (ß(u>), T) are equal to w. B. The space ( yS(to), T) is locally countable. Thus, the pseudocharacter and tightness of ( ß(u>), T) equal w. C. Every infinite closed set in (ß(ca), T) has cardinality 2C. Thus, (ß(oS), T) is countably compact, but has no nontrivial convergent sequences. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use T2-SPACES 149 D. The space (ß(u), T) has a discrete subspace Z with \Z\ =2C. Thus the weight, spread, height, and width of ( ß(ca), T) are equal to 2C. Further, every infinite subset of « has 2C cluster points in Z. E. The space (ß(u>), T) is right separated. Thus, its Lindelóf degree is 2C. We now give the definitions of the main terms which are used in this paper, and refer the reader to [5] or [10] for any omitted definitions. A space is called locally countable if its local cardinality is w. The weight (resp. density) of a space is the smallest infinite cardinal number which is the cardinality of a base (resp. dense subset) of the space. A space (A', S) is called a Urysohn space provided for every pair of distinct points x, y in X, there exist U, V in 5 such that x E U, y E V, and c\s(U) n cls(F) = 0. A space is called countably compact (resp. sequentially compact) provided that every sequence in the space has a cluster point (resp. convergent subsequence). A T2-space is called H-closed if it is a closed subset of every T2-space in which it is embedded. A point x in a space is called a Gs provided {x} is the intersection of countably many neighborhoods of x, and x is called a Hausdorff-Gs provided [x] is the intersection of countably many closed neighborhoods of x. A space P is called feebly compact if each discrete family of open sets in P is finite. A sequence (xn) of points in a space P is called a discrete sequence if for every x in P there exists a neighborhood V of x such that |{«: xn E V}\ < 1. A space P has property wD if for every discrete sequence (xn) in P there exists a subsequence (xn) and a discrete family { V¡: i < w} of open sets in P such that x E Vj if and only if / = j. Clearly a T,-space is countably compact if and only if it is feebly compact and has property wD. A space is called a Baire space if every countable intersection of open, dense sets is dense. 3. The examples. In order to simplify the notation in this section, let X denote the space (ß(u), T) given in Corollary 1.2, and let /?(«) denote the space (ß(u), S). Example (A). A countably compact, locally countable T2-space which is not D-compact for any D in /J(w)\<o. It is an open problem to construct a countably compact, locally countable, T3-space which is not 7)-compact for any D in yS(w)\w. (For the definition of 7)-compactness, see [2] or [7].) Such spaces (which are even T3i) have been constructed using extra axioms of set theory (see [11], [14], and [16]) but the problem is still open in ZFC. Example (A) shows that at least in the class of T2-spaces, we can construct such spaces within ZFC. It is known [7, Remark 2.11] that it is consistent that if P is a T3-space satisfying the properties of Example (A), then \P\ > c.